Group-Graph Reciprocal Pairs

In a 2018 paper, Cameron and Semeraro posed the problem of finding all group-graph reciprocal pairs. In this paper, we make a significant contribution to finding all such pairs. A group and graph form a reciprocal pair if they satisfy the relation $$P_{\Gamma,G}(x)=(-1)^nF_G(-x)$$ where $P_{\Gamma,G}(x)$ is the orbital chromatic polynomial of a graph $\Gamma$ and $F_G(x)$ is the cycle polynomial of a finite permutation group. We define a set of graphs to be \textit{$k$-stars} and prove that they satisfy a reciprocality relation with some group depending on $k$. These graphs are comprised of a complete graph with $k$ vertices and a further $\alpha$ `points' which are only connected to each vertex in the centre. This group is a subgroup of $S_k\times S_\alpha$, which is the automorphism group of a \textit{$k$-star} and $\alpha$ is the number of points on the star. We conjecture a list of group-graph reciprocal pairs.